Scaling limits of the Chern-Simons-Higgs energy

Matthias Kurzke, Daniel Spirn

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5 Scopus citations


We continue our study in [16] of the Gamma limit of the Abelian ChernSimonsHiggs energy Gcsh := 1/2 ∫U |∇ uε|2 + με2/4 |curl Aε - h ex|2/|uε|2 + 1/ε2 |uε|2 ( 1 - |u ε|2)2 dx on a bounded, simply connected, two-dimensional domain where ε → 0 and μ(ε) → μ ∈ [0, +∞]. Under the critical scaling, G(csh) ≈ |log ε|2 , we establish the Gamma limit when μ ∈ (0,+∞], and as a consequence, we are able to compute the first critical field H"1 = H"1(U,μ) for the nucleation of a vortex. Finally, we show failure of Gamma convergence when μ(μ) → 0 (this includes the self-dual case). The method entails estimating in certain weak topologies the Jacobian J(u(ε)) = det(∇ u(ε)) in terms of the ChernSimonsHiggs energy E(csh).

Original languageEnglish (US)
Pages (from-to)1-16
Number of pages16
JournalCommunications in Contemporary Mathematics
Issue number1
StatePublished - Feb 2008

Bibliographical note

Funding Information:
D. Spirn was supported in part by NSF grant DMS–0510121.


  • Chern-Simons-Higgs theory
  • Vortices


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